Time-stepping method

A time-stepping method approximates the evolution of an ordinary differential equation by successive updates at discrete times. It can be applied after a spatial finite element method has converted a time-dependent partial differential equation into an ordinary differential equation system. The implicit time-stepping method is one class of such methods.

An implicit time-stepping method determines a new solution value through an equation involving that same value. A small-step contraction mapping argument often establishes local solvability when the vector field is Lipschitz continuous. Examples include the Backward Euler method, implicit Runge-Kutta methods and implicit linear multistep methods. Solvability of the update is a separate issue from stability of a numerical method.